The head coach, the executive director of basketball operations/general manager, and the managing partner/governor do wield a bit of power in the organization. Let’s not get carried away.

But if you’re a 34-year old veteran of 14 NBA seasons, are you going to be more influenced by a coach, a GM, and a businessman or by a trio of ring-bearing veterans who are here to tell you that joining forces with them to play some high-level basketball is going to make for a very fulfilling experience, and might even produce another championship ring for your collection?

We have our answer.

That was indeed Rasheed Wallace sitting underneath the 1957 and 1959 championship banners at the Celtics’ HealthPoint practice facility yesterday. He was being introduced to the world as the newest member of the Boston Celtics. And this is a man who had choices once the season ended.

He tells us, in fact, there were five places where he thought he could be playing next season: San Antonio, Dallas, Cleveland, Orlando, and Boston.

Boston, he says, made the most sense.

“I felt as though this was a good fit,’’ he explains. “One thing these guys do is play defense. And they have a team scheme. The bottom line is that they play to win, and that is pretty much what swayed me to come here.’’

This is a man who knows he’s wanted. He’s not the fifth choice, the fourth choice, the third choice, or the first runner-up.

“When the season ended we looked at the free agent list, and the name that popped up right away was Rasheed Wallace,’’ says Rivers. “It was clear for our team that who he was, and how he plays, he was the perfect fit for our team.’’

Rivers, Danny Ainge, and Wyc Grousbeck were perfectly capable of delivering that message, checkbook in hand. But Danny reasoned it wouldn’t hurt to take out an insurance policy. He had three pretty satisfied employees on his payroll and he asked them if they wouldn’t mind turning back the clock a bit and acting as if they were in college (KG could pretend) and the coach was bringing in Mr. All-State from somewhere on his recruiting visit. You know, lay it on a little. Tell him how much you’d like to play with him, and don’t forget to mention how great the fans and city are, too.

“I think us being there really made a statement to him,’’ maintains Ray Allen. “A player can hear things from Danny and Doc, but he heard it from our mouths, face to face.’’

Doc likes the college recruiting analogy.

“I talk to John Calipari a lot,’’ Rivers says, “and when I told him what was happening he said, ‘Just make sure you don’t leave the campus without his commitment.’ And he never did go to another campus.’’

Ainge, the orchestrator of this whole thing, is now downplaying it.

“I don’t think that was the difference,’’ he insists. “He was on an emotional high right afterward, but he didn’t make his decision for a few days. I think it came down to who these guys are on the court. He had witnessed it from afar, and he wanted to be part of it.’’

The GM also claims there is nothing very unusual about players being part of personnel discussions, at least in Boston.

“Red [Auerbach] would talk with Larry [Bird], Kevin [McHale], and DJ [Dennis Johnson] about players,’’ Ainge says. “We would all say, ‘We don’t want to play with this guy,’ or “We’d like to play with that guy.’ ’’

Red was always ahead of the curve. The main reason he plucked Don Nelson off waivers was that Tom Heinsohn, among others, told Red they did not like to play against him.

But that was Red, and this is Boston. It’s not like that everywhere, as he knows, Doc knows, and the players know.

This organization sees the management-player relationship as a partnership.

In all candor, it’s easy for everyone to say this now, one championship later. It wasn’t all that long ago that Pierce was afraid he’d be playing out his career as the best player on a bad, or, at best, mediocre, team. Then Ainge was able to trade for Allen, and thus KG, who would never have remotely entertained a move to Boston otherwise, agreed to come here and - Voila! - a Big Three and championship No. 17.

Now Paul, KG, and Ray are senior members of the firm, and are included in policy planning.

“My belief is that players know a lot,’’ Ainge says. “They’re bright, very bright. They have great insights. I’d be stupid not to listen to Kevin, Paul, and Ray. I may not always agree, but I’ve got to listen.’’

There weren’t any disputes on this one. The Big Three wanted Rasheed. The coach, the GM, and the managing partner wanted Rasheed. And Rasheed decided that joining the Celtics was likewise in his best interest.

“He was only going to a team he thought could win,’’ Rivers says. “So I hope he’s right. I hope he’s a good talent evaluator.’’

But those other teams are all pretty good. There was really only one big thing separating them.